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I briefly skimmed the paper and don't see how you are getting this impression. Confidence intervals are—if we force the dichotomy—considered a frequentist rather than Bayesian tool. They point out that others are trying to squish a Bayesian interpretation on a frequentist tool by treating confidence intervals as though they are credible intervals, and they state this quite explicitly (p.17–18, emphasis mine):
Hmmm, yes, I suppose I was making the same mistake they were... I thought that what confidence intervals were are actually what credible intervals are.